Solution of a Two-Degree-of-Freedom Nonlinear Coupled Vibration System Based on the Multiple Scales Method

This paper takes the two-degree-of-freedom nonlinear coupled vibration system, which is widely present in engineering scenarios such as mechanical coupled structures and electromechanical systems, as the research object. Focusing on its cubic coupling nonlinear terms and the complex form of coexisting external excitation and parametric excitation, the multiple scales method is adopted to carry out dynamic solution and analysis. First, the governing equations of the system are defined, and the perturbation expansion of displacement responses ($y_1=y_{10}+\epsilon y_{11}$ , $y_2=y_{20}+\epsilon y_{21}$ ) is performed based on the small parameter $\epsilon$ . Then, the expansion is substituted into the original equations, and the coefficients of the same order of $\epsilon$ are matched to derive the solvability conditions under the combined action of 2nd-order superharmonic resonance and 1/2-order subharmonic resonance. Finally, the amplitude-frequency response equations of the system are solved, and the influence laws of key parameters such as nonlinear stiffness coefficients, excitation amplitudes, and damping ratios on the dynamic behaviors of the system are further discussed. The research results show that the multiple scales method can effectively capture the nonlinear dynamic characteristics of the two-degree-of-freedom coupled system, providing a theoretical reference for the design and stability control of related engineering structures.